 7.10: In Exercises 710, find the standard matrix for the linear transform...
 7.7.189: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.1: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.336: Use the diagonalization techniques of this chapter to find the popu...
 7.7.80: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.340: Calculate the first 12 terms of the Fibonacci sequence.
 7.7.346: In Exercises 1 and 2, determine whether the function is a linear tr...
 7.7.378: Using Mathematical Induction In Exercises 14, usemathematical induc...
 7.7.130: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.190: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.2: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.255: Characteristic Equation, Eigenvalues, and Basis InExercises 16, fin...
 7.7.337: Interpret the solutions in terms of the longterm population trends...
 7.7.81: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.341: Explain how the matrix identity can beused to generate the Fibonacc...
 7.7.347: In Exercises 1 and 2, determine whether the function is a linear tr...
 7.7.379: Using Mathematical Induction In Exercises 14, usemathematical induc...
 7.7.131: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.191: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.3: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.256: Characteristic Equation, Eigenvalues, and Basis InExercises 16, fin...
 7.7.338: Graph the solutions and over the domain
 7.7.82: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.342: Starting with show that where4. Find a matrix that diagonalizes5. D...
 7.7.348: Let be the linear transformation defined by where Find the dimensio...
 7.7.380: Using Mathematical Induction In Exercises 14, usemathematical induc...
 7.7.132: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.192: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.4: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.257: Characteristic Equation, Eigenvalues, and Basis InExercises 16, fin...
 7.7.339: Explain why the quotient approaches a limit as increases.
 7.7.83: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.343: Find a matrix that diagonalizes A.
 7.7.349: Let be the linear transformation defined by where Find (a) and (b) ...
 7.7.381: Using Mathematical Induction In Exercises 14, usemathematical induc...
 7.7.133: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.193: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.5: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.258: Characteristic Equation, Eigenvalues, and Basis InExercises 16, fin...
 7.7.84: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.344: Derive an explicit formula for the th term of the Fibonacci sequenc...
 7.7.350: Find the kernel of the linear transformation
 7.7.382: Proposing a Formula and Using MathematicalInduction In Exercises 5 ...
 7.7.134: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.194: Finding Age Distribution Vectors In Exercises 16,use the age transi...
 7.7.6: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.259: Characteristic Equation, Eigenvalues, and Basis InExercises 16, fin...
 7.7.85: Diagonalizable Matrices and Eigenvalues InExercises 16, (a) verify ...
 7.7.345: Determine the limit of as approaches infinity. Do you recognize thi...
 7.7.351: Let be the linear transformation defined by where Find a basis for ...
 7.7.383: Proposing a Formula and Using MathematicalInduction In Exercises 5 ...
 7.7.135: Determining Whether a Matrix Is Symmetric InExercises 16, determine...
 7.7.195: Find a stable age distribution vector for the agetransition matrix ...
 7.7.7: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.260: Characteristic Equation, Eigenvalues, and Basis InExercises 7 and 8...
 7.7.86: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.352: In Exercises 710, find the standard matrix for the linear transform...
 7.7.384: Using Mathematical Induction with Inequalities InExercises 7 and 8,...
 7.7.136: Proof In Exercises 710, prove that the symmetricmatrix is diagonali...
 7.7.196: Find a stable age distribution vector for the agetransition matrix ...
 7.7.8: Verifying Eigenvalues and Eigenvectors In Exercises18, verify that ...
 7.7.261: Characteristic Equation, Eigenvalues, and Basis InExercises 7 and 8...
 7.7.87: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.353: In Exercises 710, find the standard matrix for the linear transform...
 7.7.385: Using Mathematical Induction with Inequalities InExercises 7 and 8,...
 7.7.137: Proof In Exercises 710, prove that the symmetricmatrix is diagonali...
 7.7.197: Find a stable age distribution vector for the agetransition matrix ...
 7.7.9: Use and from Exercise 3 to show that(a) for any real number(b) for ...
 7.7.262: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.88: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.354: In Exercises 710, find the standard matrix for the linear transform...
 7.7.386: Using Mathematical Induction Use mathematicalinduction to prove tha...
 7.7.138: Proof In Exercises 710, prove that the symmetricmatrix is diagonali...
 7.7.198: Find a stable age distribution vector for the agetransition matrix ...
 7.7.10: Use and from Exercise 5 to show that(a) for any real number(b) for ...
 7.7.263: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.89: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.387: Using Mathematical Induction in Linear Algebra(From Chapter 2) Use ...
 7.7.139: Proof In Exercises 710, prove that the symmetricmatrix is diagonali...
 7.7.199: Find a stable age distribution vector for the agetransition matrix ...
 7.7.11: Determining Eigenvectors In Exercises 1114,determine whether is an ...
 7.7.264: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.90: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.356: Find the standard matrix for the linear transformation thatprojects...
 7.7.388: Using Mathematical Induction in Linear Algebra(From Chapter 3) Use ...
 7.7.140: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.200: Find a stable age distribution vector for the agetransition matrix ...
 7.7.12: Determining Eigenvectors In Exercises 1114,determine whether is an ...
 7.7.265: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.91: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.357: Let be the linear transformation defined by a counterclockwise rota...
 7.7.389: Using Mathematical Induction in Linear Algebra(From Chapter 6) Use ...
 7.7.141: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.201: Population Growth Model A population has thefollowing characteristi...
 7.7.13: Determining Eigenvectors In Exercises 1114,determine whether is an ...
 7.7.266: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.92: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.358: In Exercises 13 and 14, find the standard matrices for
 7.7.390: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.142: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.202: Population Growth Model A population has thefollowing characteristi...
 7.7.14: Determining Eigenvectors In Exercises 1114,determine whether is an ...
 7.7.267: Determining Whether a Matrix Is Diagonalizable InExercises 914, det...
 7.7.93: Diagonalizing a Matrix In Exercises 714, for eachmatrix find (if po...
 7.7.359: In Exercises 13 and 14, find the standard matrices for
 7.7.391: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.143: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.203: Population Growth Model A population has thefollowing characteristi...
 7.7.15: Finding Eigenspaces in Geometrically InExercises 15 and 16, use the...
 7.7.268: For what value(s) of does the matrixhave the following characterist...
 7.7.94: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.360: Find the inverse of the linear transformation defined by
 7.7.392: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.144: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.204: Find the limit (if it exists) of as approachesinfinity for the foll...
 7.7.16: Finding Eigenspaces in Geometrically InExercises 15 and 16, use the...
 7.7.269: Show that if then the transformation for acounterclockwise rotation...
 7.7.95: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.361: . Determine whether the linear transformation defined byis invertib...
 7.7.393: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.145: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.205: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.17: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.270: Writing In Exercises 1720, explain why the matrix isnot diagonaliza...
 7.7.96: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.362: Find the matrix of the linear transformation relative to thebases f...
 7.7.394: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.146: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.206: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.18: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.271: Writing In Exercises 1720, explain why the matrix isnot diagonaliza...
 7.7.97: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.363: Let and be bases for(a) Find the matrix of relative to thebasis(b) ...
 7.7.395: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.147: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.207: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.19: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.272: Writing In Exercises 1720, explain why the matrix isnot diagonaliza...
 7.7.98: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.364: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 7.7.396: Using Proof by Contradiction In Exercises 1319, useproof by contrad...
 7.7.148: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.208: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.20: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.273: Writing In Exercises 1720, explain why the matrix isnot diagonaliza...
 7.7.99: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.365: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 7.7.397: Using Proof by Contradiction in Linear Algebra(From Chapter 3) Use ...
 7.7.149: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.209: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.21: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.274: Determining Whether Two Matrices Are Similar InExercises 2124, dete...
 7.7.100: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.366: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 7.7.398: Using Proof by Contradiction in Linear Algebra(From Chapter 4) Use ...
 7.7.150: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.210: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.22: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.275: Determining Whether Two Matrices Are Similar InExercises 2124, dete...
 7.7.101: Showing That a Matrix Is Not DiagonalizableIn Exercises 1522, show ...
 7.7.367: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 7.7.399: Using Proof by Contradiction in Linear Algebra(From Chapter 4) Let ...
 7.7.151: Finding Eigenvalues and Dimensions of EigenspacesIn Exercises 1122,...
 7.7.211: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.23: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.276: Determining Whether Two Matrices Are Similar InExercises 2124, dete...
 7.7.102: Determining a Sufficient Condition for DiagonalizationIn Exercises ...
 7.7.368: In Exercises 23 and 24, find a nonsingular matrix such that is diag...
 7.7.400: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.152: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.212: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.24: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.277: Determining Whether Two Matrices Are Similar InExercises 2124, dete...
 7.7.103: Determining a Sufficient Condition for DiagonalizationIn Exercises ...
 7.7.369: In Exercises 23 and 24, find a nonsingular matrix such that is diag...
 7.7.401: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.153: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.213: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.25: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.278: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.104: Determining a Sufficient Condition for DiagonalizationIn Exercises ...
 7.7.370: Find a basis for such that the matrix forrelative to is diagonal.
 7.7.402: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.154: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.214: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.26: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.279: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.105: Determining a Sufficient Condition for DiagonalizationIn Exercises ...
 7.7.371: Find an orthogonal matrix such that diagonalizes the symmetric matrix
 7.7.403: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.155: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.215: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.27: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.280: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.106: Finding a Basis In Exercises 2730, find a basis forthe domain of su...
 7.7.372: Use the GramSchmidt orthonormalization process to find an orthogon...
 7.7.404: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.156: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.216: Solving a System of Linear Differential Equations InExercises 1728,...
 7.7.28: Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercises ...
 7.7.281: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.107: Finding a Basis In Exercises 2730, find a basis forthe domain of su...
 7.7.373: Solve the system of differential equations.
 7.7.405: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.157: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.217: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.29: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.282: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.108: Finding a Basis In Exercises 2730, find a basis forthe domain of su...
 7.7.374: Find the matrix of the quadratic form associated with the quadratic...
 7.7.406: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.158: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.218: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.30: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.283: Determining Symmetric and Orthogonal MatricesIn Exercises 2530, det...
 7.7.109: Finding a Basis In Exercises 2730, find a basis forthe domain of su...
 7.7.375: A population has the following characteristics.(a) A total of 80% o...
 7.7.407: Using a Counterexample In Exercises 2330, use acounterexample to sh...
 7.7.159: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.219: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.31: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.284: Eigenvectors of a Symmetric Matrix In Exercises3134, show that any ...
 7.7.110: roof Let be a diagonalizable matrix and letbe an invertible matrix ...
 7.7.376: Define an orthogonal matrix.
 7.7.160: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.220: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.32: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.285: Eigenvectors of a Symmetric Matrix In Exercises3134, show that any ...
 7.7.111: Let be distinct eigenvalues of thematrix Use the result of Exercise...
 7.7.377: Prove that if is similar to and is diagonalizable, then is diagonal...
 7.7.161: Determining Whether a Matrix Is Orthogonal InExercises 2332, determ...
 7.7.221: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.33: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.286: Eigenvectors of a Symmetric Matrix In Exercises3134, show that any ...
 7.7.112: Finding a Power of a Matrix In Exercises 3336, usethe result of Exe...
 7.7.162: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.222: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.34: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.287: Eigenvectors of a Symmetric Matrix In Exercises3134, show that any ...
 7.7.113: Finding a Power of a Matrix In Exercises 3336, usethe result of Exe...
 7.7.163: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.223: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.35: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.288: Orthogonally Diagonalizable Matrices In Exercises35 and 36, determi...
 7.7.114: Finding a Power of a Matrix In Exercises 3336, usethe result of Exe...
 7.7.164: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.224: Solving a System of Linear Differential Equations InExercises 2936,...
 7.7.36: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.289: Orthogonally Diagonalizable Matrices In Exercises35 and 36, determi...
 7.7.115: Finding a Power of a Matrix In Exercises 3336, usethe result of Exe...
 7.7.165: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.225: Writing a System and Verifying the General SolutionIn Exercises 374...
 7.7.37: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.290: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.116: True or False? In Exercises 37 and 38, determinewhether each statem...
 7.7.166: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.226: Writing a System and Verifying the General SolutionIn Exercises 374...
 7.7.38: Finding Eigenvalues In Exercises 2938, use asoftware program or a g...
 7.7.291: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.117: True or False? In Exercises 37 and 38, determinewhether each statem...
 7.7.167: Eigenvectors of a Symmetric Matrix In Exercises3338, show that any ...
 7.7.227: Writing a System and Verifying the General SolutionIn Exercises 374...
 7.7.39: Eigenvalues of Triangular and Diagonal Matrices InExercises 3942, f...
 7.7.292: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.118: Are the two matrices similar? If so, find a matrix suchthat
 7.7.168: Orthogonally Diagonalizable Matrices In Exercises3942, determine wh...
 7.7.228: Writing a System and Verifying the General SolutionIn Exercises 374...
 7.7.40: Eigenvalues of Triangular and Diagonal Matrices InExercises 3942, f...
 7.7.293: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.119: Calculus If is a real number, then you can defineby the series In a...
 7.7.169: Orthogonally Diagonalizable Matrices In Exercises3942, determine wh...
 7.7.229: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.41: Eigenvalues of Triangular and Diagonal Matrices InExercises 3942, f...
 7.7.294: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.120: Writing Can a matrix be similar to two differentdiagonal matrices? ...
 7.7.170: Orthogonally Diagonalizable Matrices In Exercises3942, determine wh...
 7.7.230: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.42: Eigenvalues of Triangular and Diagonal Matrices InExercises 3942, f...
 7.7.295: Orthogonal Diagonalization In Exercises 3742, finda matrix that ort...
 7.7.121: Prove that if is diagonalizable, then At isdiagonalizable
 7.7.171: Orthogonally Diagonalizable Matrices In Exercises3942, determine wh...
 7.7.231: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.43: Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises ...
 7.7.296: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.122: Proof Prove that if is diagonalizable with realeigenvalues
 7.7.172: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.232: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.44: Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises ...
 7.7.297: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.123: Proof Prove that the matrixis diagonalizable when and is notdiagona...
 7.7.173: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.233: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.45: Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises ...
 7.7.298: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.124: Guided Proof Prove that if the eigenvalues of adiagonalizable matri...
 7.7.174: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.234: Finding the Matrix of a Quadratic Form In Exercises4146, find the m...
 7.7.46: Eigenvalues and Eigenvectors of Linear TransformationsIn Exercises ...
 7.7.299: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.125: Guided Proof Prove that nonzero nilpotent matricesare not diagonali...
 7.7.175: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.235: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.47: CayleyHamilton Theorem In Exercises 4750,demonstrate the CayleyHa...
 7.7.300: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.126: Proof Prove that if is a nonsingular diagonalizablematrix, then is ...
 7.7.176: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.236: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.48: CayleyHamilton Theorem In Exercises 4750,demonstrate the CayleyHa...
 7.7.301: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.127: Explain how to determinewhether an n x n matrix a is diagonalizable...
 7.7.177: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.237: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.49: CayleyHamilton Theorem In Exercises 4750,demonstrate the CayleyHa...
 7.7.302: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.128: Showing That a Matrix Is Not Diagonalizable InExercises 49 and 50, ...
 7.7.178: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.238: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.50: CayleyHamilton Theorem In Exercises 4750,demonstrate the CayleyHa...
 7.7.303: Steady State Probability Vector In Exercises 4350,find the steady s...
 7.7.129: Showing That a Matrix Is Not Diagonalizable InExercises 49 and 50, ...
 7.7.179: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.239: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.51: Perform the following computational checks on theeigenvalues found ...
 7.7.304: Proof Prove that if is an symmetric matrix,then is symmetric for an...
 7.7.180: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.240: Finding the Matrix of a Quadratic Form In Exercises4752, find the m...
 7.7.52: Perform the following computational checks on theeigenvalues found ...
 7.7.305: Show that the characteristic equation of
 7.7.181: Orthogonal Diagonalization In Exercises 4352, finda matrix such tha...
 7.7.241: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.53: Show that if is an matrix whose th row isidentical to the th row of...
 7.7.306: Finding the Companion Matrix and Eigenvalues InExercises 53 and 54,...
 7.7.182: True or False? In Exercises 53 and 54, determinewhether each statem...
 7.7.242: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.54: . Proof Prove that is an eigenvalue of if andonly if is singular.
 7.7.307: Finding the Companion Matrix and Eigenvalues InExercises 53 and 54,...
 7.7.183: (a) A square matrix is orthogonal when it isinvertiblethat is, when...
 7.7.243: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.55: Proof For an invertible matrix prove that andhave the same eigenvec...
 7.7.308: The characteristic equation of the matrix
 7.7.184: Proof Prove that if and are orthogonalmatrices, then and are orthog...
 7.7.244: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.56: Proof Prove that and have the same eigenvalues.Are the eigenspaces ...
 7.7.309: Repeat Exercise 55 for the matrix
 7.7.185: Proof Prove that if a symmetric matrix has only oneeigenvalue then
 7.7.245: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.57: . Proof Prove that the constant term of the characteristicpolynomia...
 7.7.310: Proof Let be an matrix.(a) Prove or disprove that an eigenvector of...
 7.7.186: Proof Prove that if is an orthogonal matrix, then soare and59. Find...
 7.7.246: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.58: . Define by where is a fixedvector in Show that the eigenvalues of ...
 7.7.311: Proof Let be an matrix. Prove that ifthen is an eigenvector of What...
 7.7.187: Consider the following matrix.(a) Is symmetric? Explain.(b) Is diag...
 7.7.188: Find and for the following matrix. What doyou observe?
 7.7.247: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.59: Guided Proof Prove that a triangular matrix isnonsingular if and on...
 7.7.312: Proof Let and be matrices. Prove that ifis nonsingular, then is sim...
 7.7.248: Rotation of a Conic In Exercises 5360, use thePrincipal Axes Theore...
 7.7.60: Guided Proof Prove that if then 0 is theonly eigenvalue ofGetting S...
 7.7.313: Proof(a) Find a symmetric matrix such that for thematrix(b) General...
 7.7.249: Rotation of a Quadric Surface In Exercises 6164,find the matrix of ...
 7.7.61: Proof Prove that the multiplicity of an eigenvalue isgreater than o...
 7.7.314: Find an orthogonal matrix such that isdiagonal for the matrix
 7.7.250: Rotation of a Quadric Surface In Exercises 6164,find the matrix of ...
 7.7.62: An matrix has thecharacteristic equation(a) What are the eigenvalue...
 7.7.315: Writing Let be an idempotent matrix (that is,). Describe the eigenv...
 7.7.251: Rotation of a Quadric Surface In Exercises 6164,find the matrix of ...
 7.7.63: When the eigenvalues ofare and what are the possible values of a an...
 7.7.316: Writing The following matrix has an eigenvalueof multiplicity 4.(a)...
 7.7.252: Rotation of a Quadric Surface In Exercises 6164,find the matrix of ...
 7.7.64: Show thathas no real eigenvalues.
 7.7.317: Determine all symmetric matrices that have 0 astheir only eigenvalue.
 7.7.253: Let be a orthogonal matrix such thatShow that there exists a number...
 7.7.65: True or False? In Exercises 65 and 66, determinewhether each statem...
 7.7.318: True or False? In Exercises 65 and 66, determinewhether each statem...
 7.7.66: True or False? In Exercises 65 and 66, determinewhether each statem...
 7.7.319: (a) An eigenvalue of a matrix is a scalar such thatdet(b) An eigenv...
 7.7.67: Finding the Dimension of an Eigenspace InExercises 6770, find the d...
 7.7.320: Finding Age Distribution Vectors In Exercises 6770,use the age tran...
 7.7.68: Finding the Dimension of an Eigenspace InExercises 6770, find the d...
 7.7.321: Finding Age Distribution Vectors In Exercises 6770,use the age tran...
 7.7.69: Finding the Dimension of an Eigenspace InExercises 6770, find the d...
 7.7.322: Finding Age Distribution Vectors In Exercises 6770,use the age tran...
 7.7.70: Finding the Dimension of an Eigenspace InExercises 6770, find the d...
 7.7.323: Finding Age Distribution Vectors In Exercises 6770,use the age tran...
 7.7.71: . Calculus Let be given byShow that is an eigenvalue of with corres...
 7.7.324: Population Growth Model A population has thefollowing characteristi...
 7.7.72: Calculus For the linear transformation given inExercise 71, find th...
 7.7.325: Population Growth Model A population has thefollowing characteristi...
 7.7.73: Define byFind the eigenvalues and the eigenvectors of relativeto th...
 7.7.326: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.74: Define byFind the eigenvalues and eigenvectors of relative tothe st...
 7.7.327: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.75: Define byFind the eigenvalues and eigenvectors of relative tothe st...
 7.7.328: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.76: Find all values of the angle for which the matrix has real eigenval...
 7.7.329: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.77: . What are the possible eigenvalues of an idempotentmatrix? (Recall...
 7.7.330: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.78: What are the possible eigenvalues of a nilpotent matrix?(Recall tha...
 7.7.331: Solving a System of Linear Differential Equations InExercises 7378,...
 7.7.79: Proof Let be an matrix such that the sum ofthe entries in each row ...
 7.7.332: Rotation of a Conic In Exercises 7982, (a) find thematrix of the qu...
 7.7.333: Rotation of a Conic In Exercises 7982, (a) find thematrix of the qu...
 7.7.334: Rotation of a Conic In Exercises 7982, (a) find thematrix of the qu...
 7.7.335: Rotation of a Conic In Exercises 7982, (a) find thematrix of the qu...
Solutions for Chapter 7: Eigenvalues and Eigenvectors
Full solutions for Elementary Linear Algebra  7th Edition
ISBN: 9781133110873
Solutions for Chapter 7: Eigenvalues and Eigenvectors
Get Full SolutionsElementary Linear Algebra was written by and is associated to the ISBN: 9781133110873. Chapter 7: Eigenvalues and Eigenvectors includes 406 full stepbystep solutions. Since 406 problems in chapter 7: Eigenvalues and Eigenvectors have been answered, more than 11587 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 7.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·